1. (10 pts.) Using the formula, find the unique x between 0 and 2001 such that

x ≡ 1 (mod 2) , x ≡ 5 (mod 7) , x ≡ 6 (mod 11) , x ≡ 4 (mod 13) .

Reminder (Chinese Remainder Theorem, General Version) If n_{1}, n_{2}, …, n_{k} are pairwise relatively prime (i.e. gcd(n_{i}, n_{j}) = 1, 1 ≤ i < j ≤ k), then the unique x , 0 ≤ x < n_{1} ⋯ n_{k} satisfying

x ≡ a_{i} (mod n_{i}) , 1 ≤ i ≤ k

is (letting N = n_{1} ⋯ n_{k})

x = ∑_{i=1}^{k} a_{i} \frac{N}{n_{i}} · ((\frac{N}{n_{i}})^{−1} (mod n_{i}))

2. (10 pts.) What is the day of the week of Nov. 26, 3024. Reminders: Every year that is a multiple of 4 is a leap year, with the exception that every year that is a multiple of 100 is not a leap year, with the exception to the exception that if the year is a multiple of 400 it is a leap year.

3. a (5 pts.) What is is the remainder when you divide 102! by 103? Explain! What theorem are you using? b (5 pts.) What is is the remainder when you divide 11^{1002} by 1003? What theorem are you using?

4. a (3 pts.) Define Euler’s Toitent function ϕ(n). b (7 pts. altogether) State (2 pts.) and prove (5 pts), Euler’s Classical Formula for Euler’s Toitent function ϕ(n).

5. (10 pts.) Using the RSA encrypton method, with p = 5 , q = 11 , e = 7 , Alice sent you the encripted message c = 2 what was her original message m?

6. (10 pts., 2 pts. each) What are σ(105), σ_{2}(105), σ_{3}(105), σ_{4}(105), σ_{5}(105)?

7. (10 pts., 5 pts. each) Find (a): μ(2002) (b): μ(4004). Explain!

8. (10 pts.) Is 17 a quadratic residue modulo 281? Explain! Rule 1: If p is an odd prime and a and b are not multiples of p, then

(\frac{ab}{p}) = (\frac{a}{p}) · (\frac{b}{p})

Rule 2: If p is an odd prime then

(\frac{−1}{p}) = (−1)^{(p−1)/2} .

Rule 3: If p is an odd prime then

(\frac{2}{p}) = (−1)^{(p^{2}−1)/8} .

Rule 4: (THE QUADRATIC-RECIPROCITY LAW) If p and q are distinct odd primes, then

(\frac{p}{q}) (\frac{q}{p}) = (−1)^{(p−1)(q−1)/4} .

9. (10 pts.) For the following partitions λ, (i) Draw the Ferrers graph (ii) Find the conjugate partition λ′.

λ = (7, 5, 5, 3, 2, 1, 1, 1)

10. (10 pts. altogether)

i. (5 pts) Apply Glashier’s bijection (in the distinct → odd direction) to the distinct partition (8, 6, 3, 2, 1) to get an partition, call it μ

ii. (5 pts.) Apply Glashier’s bijection (in the odd → distinct direction) to the partition μ and show that you get (8, 6, 3, 2, 1) back, as you should.